6. Suppose that two identical firms produce widgets and that they are the only f

Question

6. Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 60Q1 and C2 = 60Q2, where Q1 is the output of Firm 1 and Q2 the output of Firm 2. Price is determined by the following demand curve:
P = 300 – Q
where Q = Q1 + Q2.
a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.
b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.
c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1’s profit differ from that found in part (b) above?
d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profits?

Explanation / Answer

a. 1 = P Q1 C1 = (300 Q1 Q2 )Q1 60Q1 = 300Q1 Q1^2 Q1 Q2 60Q1
2 = P Q2 C2 = (300 Q1 Q2 )Q2 60Q2 = 300Q2 Q1 Q2 Q2^2-60Q2
Take the FOCs:

/(Q1)= 300 2Q1 Q2 = 0 Q1 = 120 0.5Q2
/(Q2)= 300 Q1 2Q2 = 0 Q2 = 120 0.5Q1
Q1 = 120 0.5[120 0.5Q1 ] = 60 0.25Q1 Q1 = 80
Similarly nd Q2 = 80 such that 1 = 2 = 6, 400.

b. The two rms act as a monopolist, where each rm produces an equal share of total output. Demand is given by P = 300 Q, M R = 300 2Q, and M C = 60. Set M C = M R tond that Q = 120 and Q1 = Q2 = 60, respectively. Therefore:
1 = 2 = 180 × 60 60 × 60 = 7, 200.

c. It would be higher because they could make more money.

d. Firm 2 knows that Q1 = 60 and given the reaction function derived in part (a) rm 2 sets Q2 = 120 0.5 × 60 = 90. Overall, QT = 150 and P = 300 150 = 150. Hence:
1 = 150 × 60 60 × 60 = 5, 400
2 = 150 × 90 60 × 90 = 8, 100.

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